304,327 research outputs found
Group-invariant solutions of a nonlinear acoustics model
Based on a recent classification of subalgebras of the symmetry algebra of
the Zabolotskaya-Khokhlov equation, all similarity reductions of this equation
into ordinary differential equations are obtained. Large classes of
group-invariant solutions of the equation are also determined, and some
properties of the reduced equations and exact solutions are discussed.Comment: 14 page
Supervised Learning with Similarity Functions
We address the problem of general supervised learning when data can only be
accessed through an (indefinite) similarity function between data points.
Existing work on learning with indefinite kernels has concentrated solely on
binary/multi-class classification problems. We propose a model that is generic
enough to handle any supervised learning task and also subsumes the model
previously proposed for classification. We give a "goodness" criterion for
similarity functions w.r.t. a given supervised learning task and then adapt a
well-known landmarking technique to provide efficient algorithms for supervised
learning using "good" similarity functions. We demonstrate the effectiveness of
our model on three important super-vised learning problems: a) real-valued
regression, b) ordinal regression and c) ranking where we show that our method
guarantees bounded generalization error. Furthermore, for the case of
real-valued regression, we give a natural goodness definition that, when used
in conjunction with a recent result in sparse vector recovery, guarantees a
sparse predictor with bounded generalization error. Finally, we report results
of our learning algorithms on regression and ordinal regression tasks using
non-PSD similarity functions and demonstrate the effectiveness of our
algorithms, especially that of the sparse landmark selection algorithm that
achieves significantly higher accuracies than the baseline methods while
offering reduced computational costs.Comment: To appear in the proceedings of NIPS 2012, 30 page
Group properties and invariant solutions of a sixth-order thin film equation in viscous fluid
Using group theoretical methods, we analyze the generalization of a
one-dimensional sixth-order thin film equation which arises in considering the
motion of a thin film of viscous fluid driven by an overlying elastic plate.
The most general Lie group classification of point symmetries, its Lie algebra,
and the equivalence group are obtained. Similar reductions are performed and
invariant solutions are constructed. It is found that some similarity solutions
are of great physical interest such as sink and source solutions,
travelling-wave solutions, waiting-time solutions, and blow-up solutions.Comment: 8 page
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